abstract r arxiv:1609.06834v2 [hep-ph] 31 may 2017once r-parity violation is accepted, the gravitino...
TRANSCRIPT
IPMU16-0140
Revisiting gravitino dark matter in thermal leptogenesis
Masahiro Ibe,1, 2, ∗ Motoo Suzuki,1, 2, † and Tsutomu T. Yanagida1, ‡
1Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
2ICRR, The University of Tokyo, Kashiwa, Chiba 277-8582, Japan
Abstract
In this paper, we revisit the gravitino dark matter scenario in the presence of the bilinear R-
parity violating interaction. In particular, we discuss a consistency with the thermal leptogenesis.
For a high reheating temperature required for the thermal leptogenesis, the gravitino dark
matter tends to be overproduced, which puts a severe upper limit on the gluino mass. As we
will show, a large portion of parameter space of the gravitino dark matter scenario has been
excluded by combining the constraints from the gravitino abundance and the null results of the
searches for the superparticles at the LHC experiments. In particular, the models with the stau
(and other charged slepton) NLSP has been almost excluded by the searches for the long-lived
charged particles at the LHC unless the required reheating temperature is somewhat lowered by
assuming, for example, a degenerated right-handed neutrino mass spectrum.
∗ e-mail: [email protected]† e-mail: [email protected]‡ e-mail: [email protected]
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I. INTRODUCTION
For decades, supersymmetry has been widely studied as one of the top candidates for
physics beyond the Standard Model (SM) which allows a vast separation of low energy
scales from high energy scales such as the Planck scale. The precise unification of the
gauge coupling constants at the scale of the grand unified theory (GUT) also strongly
supports the minimal supersymmetric standard model (MSSM). In addition, when the
R-parity [1] is imposed to forbid baryon (B) and lepton (L) number violating interactions,
the lightest supersymmetric particle (LSP) becomes a good candidate for dark matter.
Although the R-parity is very important phenomenologically, understanding of its
origin remains as an open question [2] . In fact, in view of general discussion that all
global symmetries are necessarily broken by quantum gravity effects [3–8], the R-parity
is potentially violated unless it is embedded in gauged symmetries.1
A popular framework for such embedding is to identify the R-parity (or the mat-
ter parity [13–16]) with a discrete Z2 subgroup of the gauged U(1)B−L symmetry, which
naturally emerges once we introduce right-handed neutrinos required for the seesaw mech-
anism [17, 18] [see also 19]. There, the Majorana masses of the right-handed neutrinos are
induced when the U(1)B−L symmetry is broken down to its Z2 subgroup spontaneously.
This framework is, however, known to have a tension with perturbative SO(10) GUT.
There, the Majorana mass terms of the right-handed neutrinos require the vacuum ex-
pectation value (VEV) of fields in 126 or larger representations of SO(10). However, an
introduction of fields in such large representations causes a rapid blow up of the SO(10)
gauge coupling constant just above the GUT scale. To avoid this problem, it is often
assumed that U(1)B−L by a VEV of 16 representation with which the Majorana masses
are given by⟨16⟩2
. In this case, the Z2 subgroup of U(1)B−L does not remain unbroken,
and hence, no exact R-parity remains.2 Rather, this argument opens up a new frame-
work3 where small R-parity violation effects are tied with U(1)B−L breaking as pursued
in Refs. [21, 22].
Once R-parity violation is accepted, the gravitino LSP has a definite advantage to be
1 Discrete subgroups of the gauge symmetries are immune to quantum gravitational effects [9–12].2 See [20] for a recent discussion on R-parity violation in string theory.3 This does not preclude the R-parity originating from symmetries other than U(1)B−L though.
2
a candidate for dark matter. Compared with other LSP candidates, the gravitino LSP
can have a much longer lifetime even in the presence of R-parity violation [23, 24].
In this paper, we discuss gravitino dark matter in the presence of the R-parity violating
interactions. In particular, we revisit a consistency with the thermal leptogenesis [25] [see
26–28, for review]. For a high reheating temperature required for the thermal leptogenesis,
the gravitino dark matter tends to be overproduced, which puts a severe upper limit on
the gluino mass [21, 29, 30]. As we will show, a large portion of parameter space of the
gravitino dark matter scenario has been excluded by combining the constraints from the
gravitino abundance and the null results of the searches for the superparticles at the LHC
experiments.
The organization of this paper is as follows. In Sec. II, we briefly review R-parity
violation in the MSSM. We also review the gravitino properties in the presence of R-
parity violation. In Sec. III, we discuss a consistency between the gravitino dark matter
scenario and thermal leptogenesis scenario. There, we also discuss the constraints from
the LHC experiments. Final section is devoted to our conclusions and discussions.
II. R-PARITY VIOLATION AND THE GRAVITINO DARK MATTER
Let us briefly review R-parity violation in the MSSM (see [31] for a detailed review).
The general renormalizable R-parity violating superpotential is given by
WR =1
2λijkLLiLLjERk + λ′ijkLLiQLjDRk +
1
2λ′′ijkURiDRjDRk + µ′iLLiHu , (1)
where i, j, k = 1, 2, 3 denote the family indices of the matter fields. The coefficients λ(′,′′)
and µ′ are dimensionless and dimensionful parameters of R-parity violation, respectively.
The third term violates the B-number while the other terms violate the L-number.
The most universal constraints on R-parity violation come from cosmology. In the
presence of the B and/or L-number violating processes induced by R-parity violation, the
baryon asymmetry generated before the electroweak phase transition would be washed
out. To avoid this problem, the R-parity violating parameters are constrained to be,
λ, λ′, λ′′, µ′/µ < O(10−(6−7)) , (2)
3
where µ denotes the R-parity conserving µ-parameter in the TeV range [32–35].4 Here-
after, we suppress the family indices for simplicity.
The bilinear R-parity terms in Eq. (1) are also constrained from the neutrino mass [31].
By taking the cosmological upper limit on the neutrino mass,∑
imνi . 0.183 eV (at 95%
CL) [39], the constraint is given by,
∑i
µ′2
µ2. 2× 10−11 tan2 β
(M2
1 TeV
)(M1
M1c2W +M2s2W
). (3)
Here, M1,2 are the soft supersymmetry breaking masses of the bino and the wino, respec-
tively, tan β is the ratio between the VEVs of the two Higgs doublets, and s2W denotes the
weak mixing angle, sin2 θW , with c2W = 1− s2W . It should be noted that we take the basis
of the Higgs bosons and the sleptons, (Hd, Li), so that no sleptons obtain VEVs (see [31]
for details).
Now, let us briefly discuss R-parity violation tied to U(1)B−L breaking. In particular,
we focus on models where the effects of R-parity violation in the MSSM appear through
tiny VEVs of the right-handed sneutrinos,⟨NR
⟩[22, 40] (see also the appendix A). In
this class of models, the R-parity violating parameters are generated as5
µ′ ∼ yν
⟨NR
⟩. (4)
Thus, the constraints in Eqs. (2) and (3) on λ(′) and µ′ can be satisfied as long as 〈NR〉’sare small.
As an advantageous feature of this class of models, the B-violating term, λ′′, can be
further suppressed by additional symmetries (see the appendix A). Thus, this class of
models can evade the sever constraints from the null observation of proton decay [31] ,
|λ′λ′′| . 10−25(mSUSY
1 TeV
)2, (5)
where mSUSY denotes a typical mass of superparticles.
When the R-parity violation effects are dominated by the bilinear terms, the gravitino
4 See Ref. [36], for baryogengesis [37, 38] in the presence of the R-parity violation.5 In the basis of (Hd, Li) where no sleptons obtain VEVs, the trilinear terms are also generated as
λ ∼ λ′ ∼ µ′/µ .
4
mainly decays into a pair of a Z boson and a neutrino, a pair of a Higgs boson and a
neutrino, and a pair of a W boson and a charged lepton. The relative branching ratios
into those modes converge to 1 : 1 : 2 in the limit of m3/2 mZ,W,h [41, 42]. The decay
widths of those modes are roughly given by,
2Γ[ψ3/2 → Zν] ∼ 2Γ[ψ3/2 → hν] ∼ Γ[ψ3/2 → W`] ∼m3
3/2
192πM2PL
(µ′
µ
)2
, (6)
leading to the lifetime of the gravitino,
τ3/2 ' 1020 sec×(
1 TeV
m3/2
)3(10−7µ
µ′
)2
. (7)
Here, m3/2 denotes the gravitino mass, and MPL ' 2.4 × 1018 GeV the reduced Planck
scale. Therefore, the lifetime of the gravitino can be much longer than the age of the
universe, O(1017) sec, for µ′i/µ 10−7 for the graivitino in the hundreds GeV to a TeV
range.
The gravitino lifetime in the range of Eq. (7) is, however, severely constrained from
the observation of the extragalactic gamma-ray background (EGRB) [43–46].6 By using
50-month EGRB observation by Fermi-LAT [48], the lifetime of the gravitino dark matter
decaying into a pair of a W boson and a charged lepton is constrained to be τ3/2 & 1028 sec
for m3/2 = O(100 GeV– 1 TeV). In the following, we assume that the bilinear R-parity
violating parameters satisfy
µ′
µ. 10−11 ×
(1 TeV
m3/2
)3/2
. (8)
6 The observations of the neutrino fluxes also constraints the lifetime of the gravitino, which is less
stringent than those from the EGRB [47].
5
500 1000 1500 2000 2500 3000
2.2
2.4
2.6
2.8
3.0
mgGeV
r g
FIG. 1. The parameter rg as a function of the physical gluino mass, mg. The upper and lower
bound correspond to 10 TeV and 1 TeV squark mass. We fix β = 10 although rg barely depends
on tanβ nor other model parameters than m1/2 and the squark masses.
III. GRAVITINO DARK MATTER IN THERMAL LEPTOGENSIS SCENARIO
A. Thermal gravitino production
The productions of the gravitino from the thermal bath are dominated by the QCD
process. The resultant relic density of the gravitino dark matter is given by [49, 50],
Ω3/2h2 ' 0.09
( m3/2
100 GeV
)( TR1010 GeV
)((1 + 0.558
r−2g m2g
m23/2
)
−0.011
(1 + 3.062
r−2g m2g
m23/2
)log
[TR
1010 GeV
]), (9)
for the universal gaugino mass generated at the GUT scale.7 Here, the parameter rg is
introduced to translate the universal gaugino mass parameter, m1/2, at the GUT scale to
the physical gluino mass, mg,
mg = rgm1/2 , (10)
which depends on the MSSM parameters. In Fig.1, we show rg as a function of the
physical gluino mass calculated with the code SOFTSUSY [52] where the upper and lower
bound correspond to squark mass 10 TeV and 1 TeV. Here, we fix β = 10 although rg
7 Electroweak contributions to the gravitino production increases the abundance by around 20% when
the gaugino masses satisfy the GUT relation [51].
6
m3
2=
mb
HGU
TL
W32 h2>0.12
mg£ m32
0 200 400 600 8000
200
400
600
800
1000
1200
1400
m32GeV
mg G
eV
TR=1.4´109GeV
m3
2=
mb
HGU
TL
W32 h2>0.12
mg£ m32
0 200 400 600 800 1000 12000
500
1000
1500
2000
m32GeV
mg G
eV
TR=109GeV
FIG. 2. The upper limits on the gluino mass as a function of the gravitino mass, m3/2, for
given reheating temperatures. Below the black dashed lines, the gravitino becomes heavier than
the gluino. The gray shaded regions are excluded where the gravitino abundance exceeds the
observed dark matter abundance. The gaugino masses can satisfy the GUT relation in the left
side of the red dashed line while keeping the gravitino being the LSP.
barely depends on tan β nor other model parameters than m1/2 and the squark masses.
In our analysis, we adopt the upper bound of the Fig. 1 for a given mg, which makes
the following analysis conservative. It should be noted that the relic density in Eq. (9) is
about a factor two larger than the one in [29] which is caused by the thermal mass effects
of the gluon as discussed in [49].
From the relic density in Eq. (9), we immediately find that the gluino mass is severely
constrained from above for successful leptogenesis which requires a high reheating tem-
perature, TR & 1.4 × 109 GeV [53].8 Here, we assume that the spectrum of the right-
handed neutrinos are not degenerated. In Fig. 2, we show the upper limits on the
gluino mass for given reheating temperatures. In the figure, the gray shaded regions
are excluded where the gravitino relic density exceeds the observed dark matter density,
Ωh2 ' 0.1198± 0.0015 [54]. The dark matter density can be fully explained on the upper
limit on the gluino mass for a given gravitino mass. The figure shows that the upper
limit on the gluino mass is around 1.5 TeV for TR & 1.4× 109 GeV. In the right panel of
Fig. 2, we also show the upper limit on the gluino mass for TR ' 109 GeV in case that the
reheating temperature required for leptogenesis is somewhat relaxed.
Let us comment here that the constraints become much severer when the gaugino
8 The definitions of the reheating temperature in [26, 53] and in [50] are slightly different and the former
is about 30% larger for a given inflaton decay width.
7
masses satisfy the GUT relation,
mb : mw : mg ∼ 1 : 2 : 5 – 6 , (11)
where mb,w are the masses of the bino and the wino, respectively. When the GUT relation
is satisfied, the gravitino mass should be smaller than the bino mass
m3/2 < mb '1
5mg , (12)
so that the gravitino is the LSP. In Fig. 2, this condition can be satisfied only in the left
side of the red dashed line. There, the limits on the gluino mass are mg . 1.3 TeV for
TR > 1.4× 109 GeV and mg . 1.8 TeV for TR > 109 GeV, respectively.
In the right side of the red dashed line, on the other hand, the bino and the wino should
be heavier than the GUT relation in order for the gravitino to be the LSP. Since the heavier
bino/wino masses increase the electroweak contributions to the gravitino abundance, the
abundance in Eq. (9) underestimates the gravitino abundance in this region. Therefore,
the upper limit on the gluino mass in the right side of the red dashed line is rather
conservative.
B. NLSP contributions
As discussed in [55–57], the late-time decay of the next-to-lightest superparticle (NLSP)
also contributes to the relic gravitino density,
Ωtot3/2h
2 = Ω3/2h2 +Br3/2 ×
m3/2
mNLSP
ΩNLSPh2 . (13)
Here, ΩNLSP denotes the thermal relic density of the NLSP when it is stable, and Br3/2 the
branching fraction of the NLSP into the gravitino. In the absence of R-parity violation,
the NLSP dominantly decays into the gravitino, and hence, Br3/2 = 1. In this case, the
constraints on the gluino mass from the gravitino abundance in Fig. 2 are severer [56, 58,
59].
In the presence of R-parity violation, on the contrary, Br3/2 can be much suppressed
8
when the effects of R-parity violation are sizable. In fact, the width of the R-parity
violating decay (via the bilinear R-parity violating terms) is roughly given by,
ΓRNLSP 'κ
16π
(µ′
µ
)2
mNLSP , (14)
which leads to the lifetime,
τRNLSP ' 10−4sec× κ−1(
10−11µ
µ′
)2(1 TeV
mNLSP
). (15)
Here, κ represents dependences on the MSSM parameters [21, 41]. This width is much
larger than the width of the R-parity conserving decay into a pair of the gravitino and
the superpartner of the NLSP,
ΓRNLSP '1
48π
m5NLSP
m23/2M
2PL
, (16)
which corresponds to
τRNLSP ' 5× 103sec( m3/2
100GeV
)2( 1 TeV
mNLSP
)5
. (17)
Therefore, Br3/2 is expected to be very small even for small R-parity violating bilinear
terms as in Eq. (18).
It should be also noted that the properties of the NLSP are strongly constrained by
Big-Bang nucleosynthesis (BBN) [60, 61]. For the neutralino NLSP, for example, the
lifetime should be shorter than 102 sec to avoid dissociation of the light elements by the
NLSP decays into hadronic showers (especially into nucleons). For the stau NLSP, on
the other hand, the lifetime should be shorter than 103−4 sec to avoid the light element
dissociation. By taking those constraints from the BBN into account, we assume that the
R-parity violating parameters are in the range of,
10−14 × κ−1/2(
1 TeV
mNLSP
)1/2
.µ′
µ. 10−11 ×
(1 TeV
m3/2
)3/2
. (18)
In the appendix A, we show a model which leads to the bilinear R-parity violation terms
9
in this range. It should be noted that Br3/2 1 in this range of R-parity violation, and
hence, the NLSP contribution to the gravitino abundance in Eq. (13) is negligible.9
C. Collider Constraints
In this subsection, we discuss the constraints from the superparticle searches at the
LHC. First, let us note that the R-parity violating parameters we are interested in are
small and we can apply the search strategies for the superparticles at the LHC in the
R-conserving case.10 In fact, for the neutralino NLSP, the lifetime is typically given
by [21, 41],
cτχ01& 106 m×
(1TeV
mχ01
)3(10−11µ
µ′
)2(10
tanβ
)2
. (19)
For the stau NLSP, on the other hand, the lifetime is similarly given by [21, 41],
cττ & 107 m×(
1TeV
mτ
)(10−11µ
µ′
)2(10
tanβ
)2
. (20)
Therefore, the NLSP is stable inside the detectors in both cases.
Let us begin with the collider constraints in the case of the neutralino NLSP. Since
the neutralino NLSP is stable inside the detectors, we consider the searches for multi-jets
with missing momentum. To derive conservative limits on the gluino mass, we assume
that all the squarks are heavy and decoupled. In Fig 3, we show the constraints on the
gluino mass and the neutralino mass at the 95% CL which are extracted from the results
by the ATLAS collaboration [63].11 Here, the gluino is assumed to decay into two quarks
and a neutralino for simplicity. The figure shows that the lower limit on the gluino
mass is around 1.8 TeV when the neutralino is not degenerated with the gluino. When
the neutralino mass is close to the gluino mass, the constraints become weaker though
the the gluino mass below 1 TeV is excluded unless the neutralino is highly degenerated
with the gluino. It should be noted that the neutralino–gluino degenerated region is also
9 For typical size of ΩNLSPh2, see [56].
10 See also [62] for the effects of R-parity violation on the LHC search for much lighter gravitinos.11 See also [67], for the constraints put by the CMS collaboration.
10
m g =
m Χ
10
0 200 400 600 800 1000 12000
500
1000
1500
2000
m Χ10 GeV
mg
Ge
V
LHC constraints, neutralino NLSP
m g =
m Τ
0 200 400 600 800 1000 1200 14000
500
1000
1500
2000
mΤGeV
mg
Ge
V
LHC constraints, stau NLSP
FIG. 3. Left) The constraints on (mχ01,mg) from the searches for multi-jets with missing mo-
mentum extracted from [63]. The neutralino–gluino degenerated region is also excluded by the
mono-jet searches up to 600 GeV [64]. Right) The constraints on (mτ ,mg) from the searches
for long-lived charged particles. The constraint on the stau production cross section in [65] is
converted to the gluino mass bound by using the gluino NLO+NLL production cross section at
13 TeV (reduced by 2σ theoretical uncertainties) in [66]. The region with mτ < 340 GeV is also
excluded by the long-lived charged particle searches by assuming direct stau production [65]. It
is noted that the region below mg . 1.5 TeV with stable gluino where stau and gluino mass is
(almost) degenerate is also excluded by R-hadron search [65]. In both panels, we assume that
the squarks are heavy and decoupled. In both panels, we assume that constraints are obtained
in the limit of decoupled squarks and therefore no dependence of the squark mass is present.
excluded by the mono-jet searches up to 600 GeV [64].
For the stau NLSP, on the other hand, we consider the long-lived charged particle
searches. So far, the CMS collaboration puts a lower limit on the mass of the long-lived
stau, mτ > 340 GeV at 95% CL, by assuming a direct Drell-Yan stau pair production [65].
The CMS collaboration also puts upper limits on the production cross section of the stau
pairs for a given stau mass. Since the stau production cross section (including the one
from the cascade decays of the gluinos) depends on the gluino mass, we can put constraints
on the gluino mass for a given stau mass. In Fig. 3, we show the resultant constraints
on (mτ ,mg) plane. Here, we obtain the constraints by comparing the 95% CL limits
on the stau production cross section in [65] with the gluino production cross section in
11
FIG. 4. Combined constraints for the neutralino NLSP. The reheating temperature is assumed
to be TR = 1.4×109 GeV (left) and TR = 109 GeV (right). The gray shaded regions are excluded
where the gravitino is no more the LSP. The blue shaded regions are excluded by the missing
momentum searches [63, 67]. The GUT relation of the gaugino mass can be satisfied in the left
side of the red dashed line. The horizontal dashed lines show the upper limit on the gluino mass
for a given gravitino mass shown in Fig. 2.
[66]. The light shaded region denotes the excluded region for the central value of the
gluino NLO+NLL production cross section at 13 TeV in [66], while the darker shaded
region denotes the one for the cross section reduced by 2σ theoretical uncertainties due
to variation of the renormalization and factorization scales and the parton distribution
functions. It is noted that the region below mg . 1.5 TeV with stable gluino where stau
and gluino mass is (almost) degenerate is also excluded by the R-hadron searches which
we discuss more detail later in this section. In the following analysis, we use the later
constraint for conservative estimation.
Now, let us combine the constraints from the gravitino abundance in Fig. 2 with the
constraints in Fig. 3 from the collider searches.12 In Fig. 4, we show the constraints in
the case of the neutralino NLSP on the (mχ01,m3/2) plane. The figure shows that large
portion of the parameter region has been excluded by the LHC constraints for successful
12 In our analysis, we require that the dark matter density is dominated by the gravitino density. Hence,
we assume that the gluino mass should lie on the upper limit in Fig. 2 for a given gravitino mass.
12
FIG. 5. The same with 4 but for the stau NLSP.
leptogenesis, i.e. TR & 1.4 × 109 GeV. Even for somewhat relaxed requirement, TR &
109 GeV, some portion of the parameter region has been excluded by the LHC results.
The remaining allowed region will be tested for 300 fb−1 of integrated luminosity at 14 TeV
which reaches to mg ' 2.8 TeV [68]. If we assume the GUT relation to the gaugino masses,
the parameter region has been excluded even for somewhat lower reheating temperature
TR & 109 GeV.
In Fig. 5, we also show the combined constraints for the stau NLSP. The figure shows
that all the parameter region has been excluded by the LHC constraints for TR & 1.4 ×109 GeV. For a relaxed requirement, TR & 109 GeV, on the other hand, there remains
some allowed region, which can be also tested by further data taking.
So far, we have discussed the cases with the neutralino NLSP and the stau NLSP.
Before closing this section, let us comment on other candidates for the NLSP. When the
gluino is the NLSP, it is again stable inside the detectors, and is hadronized with the SM
quarks to form the so-called R-hadrons [69, 70]. The R-hadrons are charged unless the
gluinos are bounded with the gluons, and the charged R-hadrons can be searched for as
long-lived charged particles. So far, the CMS collaboration has excluded the gluino mass
13
below 1.5 TeV at 95% CL when the 50% of the R-hadrons are assumed to be charged [65].13
Thus, by comparing with the upper limit on the gluino mass in Fig. 2, we find that the
gravitino dark matter cannot be consistent with the thermal leptogenesis as in the case
with the stau NLSP, unless the required reheating temperature is somewhat lowered.
The CMS collaboration also puts constraints on the production cross section of the
charged R-hadron assuming a stable stop particle [65], which can be applied to the cases
of the stop NLSP and other squark NLSPs. Since the upper limits on the cross section
for a given NLSP mass are tighter than the case of the gluino NLSP, the lower limits on
the gluino mass are tighter for the squark/stop NLSP. Therefore, we again find that the
gravitino dark matter cannot be consistent with the thermal leptogenesis for these NLSP
candidates.
As for the other charged NLSPs such as selectron/smuon/charginos, the same con-
straints with the stau NLSP can be applied. For the sneutrino NLSP, on the other hand,
it leaves missing momentum inside the detector as in the case of the neutralino NLSP.
However, we need to perform more detailed analyses including model building to derive
constraints, since the event topologies depend on the decay patterns of the gluinos into
the sneutrinos. In addition, some careful parameter tunings are required to achieve the
sneutrino NLSP in the MSSM. From these points of view, we do not pursue this possibility
in this paper.
Finally, let us comment on models with a lighter gravitino. In our discussion, we have
assumed that the gravitino is in the hundreds GeV to a few TeV range, where the NLSP
decays outside of the detectors due to a limited size of R-parity violation as in Eq (8).
If the gravitino mass is a few tens of GeV, on the other hand, the bilinear R-parity
violation terms can be as large as µ′/µ ∼ 10−(7−8).14 In such cases, the NLSP lifetime
can be as short as O(10−9) sec in the bi-linear R-parity violation [21]. Thus, the neutral
NLSP leaves a displaced vertex inside the detectors, and the charged NLSP leaves a
kink inside the detectors.15 For such a light gravitino, however, the gravitino abundance
requires mg . 500 GeV which are severely constrained by the searches for multi-track
13 When the 90% of the R-hadrons are assumed to be charged, the constraints becomes mg < 1.59 TeV.14 For such a light gravitino, it mainly decays into a pair of a photon and a neutrino. The lifetime of
such a gravitino is constrained to be τ3/2 & 1029 sec by the searches for monochromatic gamma-ray
line from the Galactic center region [71].15 See e.g. [72–74] for discussions on the short lived NLSP’s.
14
displaced vertices for the neutral NLSP [75, 76] and by careful reinterpretation [77] of the
disappearing track searches for the charged NLSP [78, 79]. We leave detailed analysis for
such a light gravitino for future work.
IV. CONCLUSIONS AND DISCUSSIONS
In this paper, we revisited the gravitino dark matter scenario in the presence of the
bilinear R-parity violating interactions. In particular, we discussed the consistency with
the thermal leptogenesis. For a high reheating temperature required for the thermal
leptogenesis, the gravitino dark matter tends to be overproduced, which puts a severe
upper limit on the gluino mass. As a result, we found that a large portion of parameter
space has been excluded by the null results of the searches for multi-jets with missing
momentum at the LHC experiments when the NLSP is assumed to be the neutralino. For
the stau (and other charged slepton) NLSP, on the other hand, more stringent constraints
are put by the searches for the long-lived charged particles at the LHC experiments. As
a result, almost all the parameter space has been excluded unless the required reheating
temperature is somewhat lowered by assuming, for example, a degenerated right-handed
neutrino spectrum. For the colored NLSP candidates, constraints are tighter than the
ones for the stau NLSP, and hence, the gravitino dark matter cannot be consistent with
thermal leptogenesis in those cases, neither.
It should be noted that the constraints from cosmology are more stringent in the
absence of the R-parity violation since the late-time decay of the NLSP contributes to
the gravitino dark matter abundance [56, 58, 59]. In addition, the properties of the NLSP
are also constrained very severely by the BBN due to a long lifetime of the NLSP in
the absence of R-parity violation. As a result, the successful BBN precludes the NLSP
candidates other than the charged sleptons or the sneutrinos [56]. As for the charged
sleptons, however, the parameter region has been excluded by the LHC results as discussed
in this paper. The study of the sneutrino NLSP is a future work as mentioned above.
In our discussion, we focused on the bilinear R-parity violating interactions which
are expected to be dominant in wide range of models of spontaneous R-parity breaking
15
with the right-handed neutrinos.16 In fact, once R-parity is broken, the linear terms of
the right-handed neutrinos, εRNR, are generically allowed in the superpotential, which
leads to 〈N〉R ∼ εR/MR. Here, εR denotes an R-parity violating parameter and MR the
right-handed neutrino mass. Therefore, the resultant bilinear R-parity violating terms are
enhanced by M−1R compared with trilinear R-parity violating terms which are suppressed
not by MR but by higher cutoff scales such as the Planck scale or the GUT scale depending
on the models.
Nontheless, when R-parity violation is dominated by trilinear terms,17 the gravitino
decays into three SM fermions at a tree-level and into a pair of a SM boson and a fermion
at the one loop level [82, 83]. In those cases, the upper limits on the sizes of R-parity
violation from EGRB can be weaker than that in Eq. (8). Determination of the upper
limits on R-parity violation requires more careful analyses in those cases. If the constraints
can be relaxed to the ones in Eq. (2),18 for example, the NLSP can decay promptly inside
the detectors, which relax the LHC constraints. We leave such studies for future work.
ACKNOWLEDGEMENTS
T.T.Y. thanks Prof. Johannes Blumlein for hospitality during his stay at DESY in
Zeuthen. This work is supported in part by Grants-in-Aid for Scientific Research from the
Ministry of Education, Culture, Sports, Science, and Technology (MEXT) KAKENHI,
Japan, No. 25105011 and No. 15H05889 (M. I.) as well as No. 26104009 (T. T. Y.);
Grant-in-Aid No. 26287039 (M. I. and T. T. Y.) and No. 16H02176 (T. T. Y.) from the
Japan Society for the Promotion of Science (JSPS) KAKENHI; and by the World Premier
International Research Center Initiative (WPI), MEXT, Japan (M. I., and T. T. Y.).
16 See e.g. discussion in [80].17 See e.g. a model in [81].18 Although this seems difficult unless the gravitino mass is below a few hundred GeV, which leads to a
severer upper limits on the gluino mass as in Fig. 2.
16
Appendix A: A model of R-parity violation
1. R-parity violation tied with U(1)B−L breaking
In this appendix, we construct a model where the bilinear R-parity violating operators,
W = µ′iHuLi , (A1)
appear in the range of Eq. (18) which are appropriate for the gravitino dark matter in
the hundreds GeV to a TeV range. In particular, we consider a model where the R-
parity violation is tied to a gauged U(1)B−L breaking as motivated in SO(10) GUT
models [21, 22].
Here, we use the SO(10) GUT notation, although it is straight forward to decompose
the following discussion in terms of the MSSM fields. In the SO(10) GUT models, the
quarks and leptons are grouped into 16 representation, 16M , in conjunction with the
right-handed neutrinos. The Higgs doublets are, on the other hand, grouped into 10
representation, 10H . In our discussion, we do not specify the mechanisms which explain
the doublet-triplet splitting of the Higgs multiplets, and we assume that only the Higgs
doublets in 10H remain below the GUT scale.
In this notation, the MSSM Yukawa couplings are given by,
WMSSM = 10H16M16M , (A2)
where we have suppressed the coefficient and the family indices for simplicity. To give a
large Majorana masses to the right-handed neutrinos, we need to introduce bilinear terms
of 16M , which require at least a VEV of 126 representation. However, an introduction of
a field in the 126 representation leads to a blow up of the SO(10) gauge coupling constant
just above the GUT scale. To avoid this problem, we instead break the U(1)B−L ⊂ SO(10)
by a VEV of 16H ,
⟨16H
⟩= vB−L . (A3)
Here, as in the case of the Higgs doublets, we again assume that only the MSSM singlet
17
16M 10H 16H 16H vB−L X m3/2
R 1 0 −1/2 0 −1/4 5/2 2
TABLE I. R-charges of matter fields, Higgs fields and SO(10) singlets.
in 16H . With the VEV of 16H , the Majorana mass terms are generated from,
WNR=
1
2MPL
16H16H16M16M . (A4)
It should be noted no Z2 subgroup remains unbroken after U(1)B−L breaking.
For a later purpose, we assume that vB−L = O(10−(3−4)) × MPL, so that the right
handed neutrino masses are in the range of
MR ∼1
MPL
v2B−L ∼ 10−(6−8) ×MPL . (A5)
Then, we aim to construct a model where the right-handed sneutrinos obtain VEVs
⟨NR
⟩'v3B−LM3
PL
×m3/2 , (A6)
while the R-parity conserving µ-term is given by µ ∼ m3/2. Once these are achieved, we
obtain an appropriate bilinear R-parity violating terms,
µ′
µ∼v3B−LM3
PL
. (A7)
After the example of the model in [21], let us interconnect the R-parity violation to the
U(1)B−L breaking scale. First, in order to give a VEV to 16H , we consider a superpotential
W = 10H16M16M + 16H16H16M16M +X(16H16H − v2B−L) , (A8)
where 16H is a newly introduced 16 representation and X is an SO(10) singlet. Hereafter,
we take the unit of MPL = 1. The first term of Eq. (A8) again denotes the MSSM
Yukawa interaction, and the second term the Majorana mass term of the right-handed
neutrinos. By the assumption of the sprit multiplet, 16H and 16H contain the MSSM
18
singlets only which are absorbed into U(1)B−L gauge multiplet once they obtain the
vacuum expectation value.
In order to avoid too large R-parity violations, we forbid the following operators
W = 16M16H ,
W = 10H16M16H ,
W = 16H16M16M16M . (A9)
For that purpose, we consider a continuos R-symmetry broken by a spurion vB−L (see
more discussions in the next subsection). In Tab. A 1, we show the R-charge assignment
which forbids the above operators.
A notable feature of the R-charge assignment in Tab. A 1 is that it allows a Kahler
potential,
K = v4B−L16M16H . (A10)
which leads to a linear term of the right-handed neutrinos19
W ∼ m3/2v5B−LNR . (A11)
Thus, by combined with the Majorana mass term, the right-handed sneutrinos obtain
VEVs, ⟨NR
⟩' v3B−L ×m3/2 , (A12)
which generate the bilinear R-parity violation through the first term of Eq. (A8),
µ′ = v3B−Lm3/2 . (A13)
The R-symmetric µ-term is, on the other hand, given by
W ∼ m3/210H10H . (A14)
19 If we regard m3/2, we may directly write down this term.
19
16M 10H 16H 16H Q Q X
R 1 0 −1/2 0 −1/4 −1/4 5/2
Z10R −4 0 2 0 1 1 0
TABLE II. R-charges of matter fields, Higgs fields and SO(10) singlets. We also show the
charge assignment of Z10R with which the terms with charges −8 (mod 10) are allowed in the
superpotential.
Therefore, we find that the bilinear R-parity violation are given by,20
µ′
µ∼ v3B−L . (A15)
2. Model with discrete R-symmetry
In the above example, we made use of a continuous R-symmetry which is broken
by the spurion vB−L. In this subsection, we discuss a model where vB−L is dynamical.
For that purpose, we consider SU(5) gauge theory with four-flavor of vector-like pairs
of fundamental representation (Q, Q), and replace v2B−L to a composite operator (QQ).
Then, the superpotential in Eq. (A8) is rewritten by,
W = 10H16M16M + 16H16H16M16M +X(16H16H −QQ) , (A16)
where the R-charge assignment is given in Tab. A 2.21 Since the SU(5) gauge theory with
four-flavor does not have a vacuum, we add explicit mass terms
W = 10H16M16M + 16H16H16M16M +X(16H16H −QQ) +mQQQ . (A17)
With the explicit mass term, the R-symmetry is explicitly broken down to a discrete Z10R
symmetry whose charge assignment is given in the second line of Tab. A 2.22
20 The actual R-parity violating bilinear terms are multiplied by the neutrino Yukawa couplings.21 This charge assignment is free from the SU(5) anomaly.22 As we will see, we require mQ 1. For that purpose, we assume that mQ, X, 16H and QQ are
charged under some additional discrete symmetry.
20
Below the dynamical scale of SU(5), non-perturbative potential is generated [84]
W = 10H16M16M + 16H16H16M16M +X(16H16H −QQ)
+mQQQ+Λ11
detQQ, (A18)
where Λ denotes the dynamical scale of SU(5). As a result, QQ obtains a VEV, which
provides the spurion in the previous section
v2B−L ∼⟨QQ⟩∼(
Λ11
mQ
)1/5
. (A19)
Thus, by arranging (Λ11
mQ
)1/5
= O(10−6) , (A20)
we can provide an appropriate suprion vB−L = O(10−3). Furthermore, we can also provide
an appropriate size of the gravition mass by taking
mQ ' 10−9 , Λ ' 10−3.5 , (A21)
which leads to an appropriate VEV of the superpotential simultaneously
m3/2 = 〈W 〉 ∼ mQ
⟨QQ⟩' 10−15 . (A22)
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